Math Problem Statement

Solve using operation of function

Find (g • f)(-4) where g(x) = x + 2 and h(x) = x^3 - 2x.

Solution

The problem asks to find (gf)(4)(g \circ f)(-4), which represents the composition of two functions, g(x)g(x) and f(x)f(x).

From the image, we see that:

  • g(x)=x+2g(x) = x + 2
  • h(x)=x32xh(x) = x^3 - 2x

But it's unclear whether you're meant to compose gg with f(x)f(x) or h(x)h(x). I will assume that f(x)=h(x)f(x) = h(x), and thus f(x)=x32xf(x) = x^3 - 2x. If this assumption is incorrect, feel free to clarify!

Now, let's proceed step by step:

  1. The composition of functions means evaluating g(f(x))g(f(x)). That is, we first evaluate f(x)f(x), and then substitute the result into g(x)g(x).
  2. We need to find (gf)(4)(g \circ f)(-4), which means evaluating f(4)f(-4) and then using that value in g(x)g(x).

Let's calculate that:

  • First, evaluate f(4)f(-4):
f(-4) = (-4)^3 - 2(-4) = -64 + 8 = -56$$ - Now, use this result in $$g(x)$$: $$g(x) = x + 2 g(-56) = -56 + 2 = -54$$ Therefore, $$(g \circ f)(-4) = -54$$. Do you have any questions or would you like a deeper explanation? Here are some related questions you might find helpful: 1. What are function compositions and how do they work in general? 2. What if $$h(x)$$ was not the same as $$f(x)$$? 3. Can we generalize the process for any two functions $$g(x)$$ and $$f(x)$$? 4. What are the practical applications of composing functions? 5. How would this composition change if $$g(x)$$ were a more complex function? **Tip:** When dealing with function compositions, always evaluate the inner function first before substituting into the outer function.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Polynomial Functions

Formulas

(g ∘ f)(x) = g(f(x))

Theorems

Function Composition Theorem

Suitable Grade Level

Grades 10-12